Question

我在Ubuntu 18.04上使用fenics。

我正在尝试调整此处描述的内容：https://fenicsproject.org/pub/tutorial/sphinx1/._ftut1005.html#setting-multiple-dirichlet-neumann-and-robin-conditions

我成功地在矩形区域中做到了这一点，并获得了纽曼条件$ \ frac {\ left {\ partial u \ right}} {\ partial n} = h $ 在顶部，而Dirichlet在边界的其余部分以$ u = g $为条件。

现在我正在尝试使用相同的矩形域，顶部使用相同的Newmann条件，而其余部分使用Robin的条件：$ \ alpha u + \ beta \ frac {\ partial u} {\ partial n} = h $

首先，我尝试使用$ g = h = 0 $ 0和f dirac的增量。但是出现了这个错误：

/usr/local/lib/python3.6/dist-packages/matplotlib/colors.py:504: RuntimeWarning: invalid value encountered in less
  xa[xa < 0] = -1

这是我的代码：

from fenics import *
from dolfin import *
import matplotlib.pyplot as plt
# plot = lambda *args, **kwargs: None
import numpy as np                                      

#parameters
x0 = 0.0#Constant(0)
xN = 8.0#Constant(8)
y0 = -4.0#Constant(-4)
yM = 0.0#Constant(0)
nx = 50#Constant(10)
ny = 15#Constant(10)
hx = (xN-x0)/(nx-1)
hy = (yM-y0)/(ny-1)

# fuente y salida
in_x = 4.0
in_y = 0.0
out_x = 7.0
out_y = 0.0

d=1

#mesh
mesh = RectangleMesh(Point(x0,y0), Point(xN,yM), nx, ny, "right")
boundary_markers = MeshFunction('size_t', mesh, mesh.topology().dim()-1)
V = FunctionSpace(mesh, 'Lagrange', d)

#boundaries
class BoundaryX0(SubDomain):
    tol = 1E-14
    def inside(self, x, on_boundary):
        return on_boundary and near(x[0], x0, self.tol)

class BoundaryXN(SubDomain):
    tol = 1E-14
    def inside(self, x, on_boundary):
        return on_boundary and near(x[0], xN, self.tol)

class BoundaryY0(SubDomain):
    tol = 1E-14
    def inside(self, x, on_boundary):
        return on_boundary and near(x[1], y0, self.tol)

class BoundaryYM(SubDomain):
    tol = 1E-14
    def inside(self, x, on_boundary):
        return on_boundary and near(x[1], yM, self.tol)

bx0 = BoundaryX0()
bx0.mark(boundary_markers, 4)
bx1 = BoundaryXN()
bx1.mark(boundary_markers, 1)
by0 = BoundaryY0()
by0.mark(boundary_markers, 2)
byM = BoundaryYM()
byM.mark(boundary_markers, 3)

# dirichlet
h = Constant(0) #Expression("100*cos(x[0]*x[1])", degree=d) #Constant(0)
# newmann
g = Constant(0) #Expression("cos(x[0])", degree=1) #Constant(0) 

boundary_conditions = {4: {'Robin': h},
                       1: {'Robin': h},
                       2: {'Robin': h},
                       3: {'Neumann': g}}

ds = Measure('ds', domain=mesh, subdomain_data=boundary_markers)

# Define variational problem
u = TrialFunction(V)
v = TestFunction(V)

p = Expression("0.015+0.005*cos(3.5+x[1])", degree=d) #Constant(0.01) # #
I = 1.0
DV = hx*hy
M = I/DV
alpha = Expression("x[0]/(x[0]*x[0]+x[1]*x[1])", degree=d)
beta = Constant(1.0)

# source
f =  Constant(0) #Expression("cos(x[0]+x[1])", degree=d) #Constant(0) #source term

integrals_R_L = []
integrals_R_a = []
for i in boundary_conditions:
    if 'Robin' in boundary_conditions[i]:
        if boundary_conditions[i]['Robin'] != 0:
            hi = boundary_conditions[i]['Robin']
            integrals_R_L.append((p*hi*v/beta)*ds(i))
            integrals_R_a.append((alpha*p*u*v/beta)*ds(i))

a = p*inner(nabla_grad(u), nabla_grad(v))*dx + sum(integrals_R_a)
L = f*v*dx + p*g*v*ds(3) + sum(integrals_R_L)

# Compute solution
A, b = assemble_system(a, L)

# Add delta
delta1 = PointSource(V, Point(in_x, in_y), M)
delta1.apply(b)

# delta2 = PointSource(V, Point(out_x, out_y), -M)
# delta2.apply(b)

u = Function(V)
solve(A, u.vector(), b)

#plot y=k
y = 0.0
meshx = IntervalMesh(nx, x0, xN)
W = FunctionSpace(meshx,"Lagrange",d)
w = Function(W)
coor = meshx.coordinates()
w_array = w.vector()#.array()

if meshx.num_vertices() == len(w_array):
    for i in range(meshx.num_vertices()):
        w_array[i] = u(coor[i][0],y)

w.vector().set_local(w_array)                                       

# plots
plt.subplot(2,3,1)
plot(f, mesh=mesh, title = "f: source term")
# plt.savefig('vc_poisson/f.png')

plt.subplot(2,3,2)
plot(h, mesh=mesh, title = "h: Dirichlet boundary condition")

plt.subplot(2,3,3)
plot(g, mesh=mesh, title = "g: Newmman boundary condition")
# plt.savefig('vc_poisson/g.png')

plt.subplot(2,3,4)
plot(p, mesh=mesh, title = "\sigma: resistivity distribution")
# plt.savefig('vc_poisson/g.png')

plt.subplot(2,3,5)
plot(u, title = "numerical solution")
# plt.savefig('vc_poisson/sol.png')

# plt.subplot(2,3,5)
# plot(mesh)
# plot(u, interactive=True, title = "numerical solution + mesh")

plt.subplot(2,3,6)
plot(w, title="numerical solution u(x,0)")
plt.savefig('vc_poisson/solution.png')

plt.figure()
plot(u.root_node(), mode='warp', title="numerical solution surface")
plt.savefig('vc_poisson/solution_surface.png')

# Save solution to file in VTK format
vtkfile = File('vc_poisson/solution.pvd')
vtkfile << u

plt.show()

错误似乎出在运算符a的罗宾边界sum(integrals_R_a)上的积分上。如果我对此发表评论，那么代码会运行，但显然解决方案不是我期望的。我必须补充说，我已经2周大了。

那么，我在做什么错了？

罗宾·纽曼边界条件的

0 个答案: